[Paper Review] On curves intersecting at most once, II

Joshua Evan Greene|arXiv (Cornell University)|Nov 4, 2018

Advanced Topology and Set Theory3 references4 citations

TL;DR

This paper establishes an upper bound of $\lesssim_k g^{k+1}\log g$ for the size of a $k$-system of simple, non-homotopic loops on a closed orientable surface of genus $g$, matching the best-known constructions up to a $\log g$ factor. The proof combines probabilistic covering space techniques with a LERF property of surface groups to reduce the problem to arc systems, leveraging a refined lifting argument to bound intersections and achieve the near-optimal bound.

ABSTRACT

We prove that on a closed, orientable surface of genus $g$, a set of simple loops with the property that no two are homotopic or intersect in more than $k$ points has cardinality $\lesssim_k g^{k+1} \log g$. The bound matches the size of the largest known construction to within a factor of $\sim_k \log g$. It generalizes an earlier result of the author, which treated the case $k=1$. The proof blends probabilistic ideas with covering space arguments related to the fact that surface groups are LERF.

Motivation & Objective

• To determine the maximum possible size of a $k$-system—set of simple, non-homotopic loops intersecting in at most $k$ points—on a closed orientable surface of genus $g$.
• To close the gap between the best known constructions and upper bounds for $k$-systems of loops, which had previously been off by a factor of $g^{2k-2}$.
• To extend the author's prior result for $k=1$ to general $k \geq 1$ using novel covering space and probabilistic techniques.
• To resolve the asymptotic growth rate of $k$-systems up to a $\log g$ factor, approaching the conjectured optimal $\sim_k g^{k+1}$.

Proposed method

• Lift loops to finite-sheeted covers of the surface using the LERF property of surface groups, ensuring that intersections lift to exactly one point with positive probability.
• Apply a probabilistic averaging argument to find a cover where a positive proportion of intersecting loop pairs lift to transverse pairs intersecting exactly once.
• Use a refined version of a prior result on arc systems (Przytycki, 2015) to bound the number of lifts intersecting a single loop in the cover.
• Combine the lifting result with a known probabilistic decomposition from prior work (Greene, 2018) to derive a global upper bound on the size of the $k$-system.
• Use the structure of the covering space to reduce the problem of counting loop intersections to counting arc intersections on a cut surface.
• Apply Theorem 8 (a covering space lifting result with uniform probability bounds) to ensure that a large fraction of pairs lift with controlled intersection number.

Experimental results

Research questions

• RQ1 What is the maximum size of a $k$-system of simple, non-homotopic loops on a closed orientable surface of genus $g$?
• RQ2 Can the upper bound for $k$-systems of loops be improved to match the size of known constructions, up to a $\log g$ factor?
• RQ3 Does the method of lifting to covers with controlled intersection behavior yield a general bound for arbitrary $k$?
• RQ4 How does the genus of the surface affect the maximum size of a $k$-system, especially when $k$ is odd?
• RQ5 Can the construction of $k$-systems be generalized beyond even $k$ to include odd $k$, and what genus constraints arise?

Key findings

• The paper establishes an upper bound of $|\Gamma| \lesssim_k g^{k+1}\log g$ for the size of any $k$-system $\Gamma$ of simple loops on a genus $g$ surface.
• The bound matches the size of the largest known constructions—specifically, $\sim_k g^{k+1}$—up to a $\log g$ factor.
• A new covering space technique using probabilistic lifting ensures that a positive proportion of intersecting pairs lift to transverse pairs intersecting exactly once.
• The method proves that the number of loops intersecting any fixed loop $\alpha \in \Gamma$ is $\lesssim_k g^{k+1}$, which is tight up to the implied constant.
• The authors construct a $k$-system of size $\sim_k g^{k+1}$ for all $k$, including odd $k$, under mild genus constraints, confirming the conjectured growth rate up to the $\log g$ factor.
• The result resolves the asymptotic growth of $k$-systems on surfaces up to a $\log g$ factor, with the bound in Theorem 2 being tight in comparison to the construction.